Let $\mathbb H : = \{z \in \mathbb C\ |\ \mathfrak {I} (z) \gt 0 \}.$ Then what will be $\partial \mathbb H\ $?
I feel like it is $\mathbb R$ but in my book I found that it is $\mathbb R_{\infty}$ which might be thought of as the one point compactification of $\mathbb R.$ If we are in the extended complex plane then it's fine but if we aren't there how can we even talk of the point at infinity. This fact confuses me a lot. Could someone shed some light on it?
Thanks for investing your valuable time in reading my question.
Yes, as you suspect, and as @Martin R comments, there is a context-dependence. Literally, in the set $\mathbb C$, the boundary is certainly just $\mathbb R$.
Many complex analysis sources are (perhaps reasonably) a little vague about whether they implicitly include "the point at infinity" or not. And, indeed, it certainly depends what's happening. Indeed, every entire function "extends" to a function on the Riemann sphere with possibly an essential singularity at $\infty$. This doesn't add much to what we already know.
In contrast, the linear fractional /Moebius transformations only really make sense on the Riemann sphere, since they typically map a point in the "finite part of $\mathbb C$" to $\infty$, and vice-versa. Nevertheless, some traditional discussions ignore this "flaw", and behave as though linear fractional transformations map $\mathbb C$ to itself. Well, yeah, actually, they approximately do. :)
(Some questions about conformal mapping of non-compact regions are also ambiguous, depending whether we intend to pay intention to $\infty$ or not.)
This sort of textbook/quiz question is yet another one of those where we have to either be able to guess the mental state of the asker, or answer in a qualified fashion. The latter is more real-life (assuming some baseline sanity about one's personal version of it): "The answer depends on context. The literal topological boundary in $\mathbb C$ is just $\mathbb R$. But if we are implicitly thinking of the upper half-plane as a subset of the Riemann sphere, then the topological boundary is $\mathbb R\cup\infty$."
(Yes, I know, in some quasi-adversarial contexts, that sort of answer is not allowed, etc., but don't take it to heart...)